The support points found a branch point nobody named, and cluster at it by 1.52

At its defaults it draws the support points found a branch point nobody named, and cluster at it by 1.52. Approximating γ√(λ + c) on an interval reaching to within 0.03 of its branch point, by a barycentric rational whose support points are chosen greedily: at each step the sample where the current approximation is worst becomes the next support point. The filled curve is the error against degree; the open one is the same barycentric form with support points spread evenly along the interval, which is the choice anybody makes who has no reason to make another. At degree 10 the adaptive rule is 702.3× better, and the advantage grows with the degree. Nothing in the algorithm is told that the function has a branch point or where it is: the points it selects lie between 0.0962 and 9.53 from it, a span of 99.06, with a mean ratio of 1.519 between consecutive distances — the geometric clustering the hand-built approximant in this field is given as a rule, arrived at from the residual.

aaa-support is one function in lib/figures/graphfit.js — three approximations the data chose the shape of — support points, a restricted perturbation, and a model made of measurements. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

Approximating γ√(λ + c) on an interval reaching to within 0.03 of its branch point, by a barycentric rational whose support points are chosen greedily: at each step the sample where the current approximation is worst becomes the next support point. The filled curve is the error against degree; the open one is the same barycentric form with support points spread evenly along the interval, which is the choice anybody makes who has no reason to make another. At degree 10 the adaptive rule is 702.3× better, and the advantage grows with the degree. Nothing in the algorithm is told that the function has a branch point or where it is: the points it selects lie between 0.0962 and 9.53 from it, a span of 99.06, with a mean ratio of 1.519 between consecutive distances — the geometric clustering the hand-built approximant in this field is given as a rule, arrived at from the residual.

Approximating γ√(λ + c) on an interval reaching to within 0.03 of its branch point, by a barycentric rational whose support points are chosen greedily: at each step the sample where the current approximation is worst becomes the next support point. The filled curve is the error against degree; the open one is the same barycentric form with support points spread evenly along the interval, which is the choice anybody makes who has no reason to make another. At degree 10 the adaptive rule is 702.3× better, and the advantage grows with the degree. Nothing in the algorithm is told that the function has a branch point or where it is: the points it selects lie between 0.0962 and 9.53 from it, a span of 99.06, with a mean ratio of 1.519 between consecutive distances — the geometric clustering the hand-built approximant in this field is given as a rule, arrived at from the residual.

Approximating γ√(λ + c) on an interval reaching to within 0.15 of its branch point, by a barycentric rational whose support points are chosen greedily: at each step the sample where the current approximation is worst becomes the next support point. The filled curve is the error against degree; the open one is the same barycentric form with support points spread evenly along the interval, which is the choice anybody makes who has no reason to make another. At degree 10 the adaptive rule is 566.4× better, and the advantage grows with the degree. Nothing in the algorithm is told that the function has a branch point or where it is: the points it selects lie between 0.481 and 9.53 from it, a span of 19.81, with a mean ratio of 1.348 between consecutive distances — the geometric clustering the hand-built approximant in this field is given as a rule, arrived at from the residual.

Approximating γ√(λ + c) on an interval reaching to within 0.05 of its branch point, by a barycentric rational whose support points are chosen greedily: at each step the sample where the current approximation is worst becomes the next support point. The filled curve is the error against degree; the open one is the same barycentric form with support points spread evenly along the interval, which is the choice anybody makes who has no reason to make another. At degree 10 the adaptive rule is 616.3× better, and the advantage grows with the degree. Nothing in the algorithm is told that the function has a branch point or where it is: the points it selects lie between 0.1603 and 9.53 from it, a span of 59.44, with a mean ratio of 1.45 between consecutive distances — the geometric clustering the hand-built approximant in this field is given as a rule, arrived at from the residual.

Approximating γ√(λ + c) on an interval reaching to within 0.02 of its branch point, by a barycentric rational whose support points are chosen greedily: at each step the sample where the current approximation is worst becomes the next support point. The filled curve is the error against degree; the open one is the same barycentric form with support points spread evenly along the interval, which is the choice anybody makes who has no reason to make another. At degree 10 the adaptive rule is 747.2× better, and the advantage grows with the degree. Nothing in the algorithm is told that the function has a branch point or where it is: the points it selects lie between 0.06413 and 9.53 from it, a span of 148.6, with a mean ratio of 1.576 between consecutive distances — the geometric clustering the hand-built approximant in this field is given as a rule, arrived at from the residual.

Approximating γ√(λ + c) on an interval reaching to within 0.008 of its branch point, by a barycentric rational whose support points are chosen greedily: at each step the sample where the current approximation is worst becomes the next support point. The filled curve is the error against degree; the open one is the same barycentric form with support points spread evenly along the interval, which is the choice anybody makes who has no reason to make another. At degree 10 the adaptive rule is 812.2× better, and the advantage grows with the degree. Nothing in the algorithm is told that the function has a branch point or where it is: the points it selects lie between 0.02565 and 9.53 from it, a span of 371.5, with a mean ratio of 1.712 between consecutive distances — the geometric clustering the hand-built approximant in this field is given as a rule, arrived at from the residual.

Every figure above checked its own claims on the way to being drawn, and a claim that failed would have stopped the picture rather than shipped a wrong one. Those checks used to leave no trace at all: a passing one returned true and the only evidence the figure had checked anything was that nothing crashed. The list below is what they actually said, collected by running this generator with an observer installed — not a description of what it is believed to check.

14 distinct claims across 6 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.

It draws a decomposition and prints its residual. It calls aaa, evenSupport, and every figure above carries the badge — which residualcheck verifies by looking for it in the emitted SVG rather than by finding the call that builds one. A badge that is constructed and then left out of the body is the failure that check exists for.

Across the library: the rule bites on 217 of 397 generators — 199 print a residual and 18 are exempt with a published reason; 180 factorise nothing. Read from lib/residual-rule.js, which is the same body the gate enforces from, and the gate's last check fails the build if this page and it disagree about any generator.